Harry Houdini was a Hungarian-born American illusionist and stunt performer, noted for his sensational escape acts. He first attracted notice in vaudeville in the US and then as "Harry 'Handcuff' Houdini" on a tour of Europe, where he challenged police forces to keep him locked up.
Erik Weisz was Harry's birth name. He named himself "Houdini" sometimes after reading the autobiography of deceased magician Jean Eugène Robert-Houdin in 1890 when he was 16 years old.
Since "Harry" was a child when the Riddles and Brad met him ("Wisconsin in 1881", he would have been approximately seven years old), he would not have read Jean's biography yet.
He doesn't actually refer to himself as Houdini during the episode. When they arrive, Xavier mentions the name but out of hearing range. Yadina later says "you can do it Houdini!" while closer to him, but Erik might not have heard it since he was focused on keeping his balance. She later says "nice moves Houdini" and "way to go Harry!" and he should've been able to hear that, but doesn't contradict them.
In the show's timeline (differing from reality) it appears that Jean was born later (he looks young) and thus was still alive, and Harry actually met him and named himself after Jean over a decade earlier than he did.
"Thirty-one years later" would be in 1912 (not necessarily still in Wisconsin) when Harry would've been approximately 38 years old. His appearance doesn't appear to have changed.
Harry is depicted with fair skin and wavy brown hair. He wears a black tuxedo, grey shoes, and a red bow tie. He has a small white cravat, and a grey shirt with 3 black buttons. His cuffs are white with yellow buttons.
He mentions "my mom says I'm always surprising people", which is a reference to his mother Cecília Steiner.